Bending and Stretching of Certain Types of Heterogeneous Aeolotropic Elastic Plates

1961 ◽  
Vol 28 (3) ◽  
pp. 402-408 ◽  
Author(s):  
E. Reissner ◽  
Y. Stavsky
Keyword(s):  
The Other ◽  
Explicit Solutions ◽  
Elastic Plates ◽  
Important Special Case ◽  
Elastic Symmetry ◽  
Transverse Bending ◽  
Special Case ◽  
Equal Thickness

The class of plates with which this paper is concerned includes as an important special case plates consisting of two orthotropic sheets of equal thickness which are laminated in such a way that the axes of elastic symmetry enclose an angle +θ with the x, y-axes in one sheet and an angle −θ in the other sheet. For plates of this type there occurs a coupling phenomenon between in-plane stretching and transverse bending which does not occur in the theory of homogeneous plates and which has not been considered in earlier work for such plates. The general results of the present paper are illustrated by means of explicit solutions for two specific plate problems.

New integral representations of the polylogarithm function

2006 ◽  
Vol 463 (2080) ◽  
pp. 897-905 ◽  
Author(s):  
Djurdje Cvijović
Keyword(s):  
Bernoulli Polynomials ◽  
Integral Representations ◽  
The Other ◽  
Important Special Case ◽  
Polylogarithm Function ◽  
Physical Problems ◽  
The Riemann Zeta Function ◽  
Dilogarithm Function ◽  
Riemann Zeta ◽  
Special Case

Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function Li s ( z ). The polylogarithm function appears in several fields of mathematics and in many physical problems. We, by making use of elementary arguments, deduce several new integral representations of the polylogarithm Li s ( z ) for any complex z for which | z |<1. Two are valid for all complex s , whenever Re  s >1. The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter s is a positive integer. Our earlier established results on the integral representations for the Riemann zeta function ζ (2 n +1), n ∈ N , follow directly as corollaries of these representations.

scholarly journals Devising methods for planning a multifactorial multilevel experiment with high dimensionality

2021 ◽  
Vol 5 (4 (113)) ◽  
pp. 64-72
Author(s):  
Lev Raskin ◽  
Oksana Sira
Keyword(s):  
Original Problem ◽  
Accurate Solution ◽  
The Other ◽  
High Dimensionality ◽  
Optimal Plan ◽  
Approximate Methods ◽  
Important Special Case ◽  
Decomposition Procedure ◽  
Relevant Task ◽  
Special Case

This paper considers the task of planning a multifactorial multilevel experiment for problems with high dimensionality. Planning an experiment is a combinatorial task. At the same time, the catastrophically rapid growth in the number of possible variants of experiment plans with an increase in the dimensionality of the problem excludes the possibility of solving it using accurate algorithms. On the other hand, approximate methods of finding the optimal plan have fundamental drawbacks. Of these, the main one is the lack of the capability to assess the proximity of the resulting solution to the optimal one. In these circumstances, searching for methods to obtain an accurate solution to the problem remains a relevant task. Two different approaches to obtaining the optimal plan for a multifactorial multilevel experiment have been considered. The first of these is based on the idea of decomposition. In this case, the initial problem with high dimensionality is reduced to a sequence of problems of smaller dimensionality, solving each of which is possible by using precise algorithms. The decomposition procedure, which is usually implemented empirically, in the considered problem of planning the experiment is solved by employing a strictly formally justified technique. The exact solutions to the problems obtained during the decomposition are combined into the desired solution to the original problem. The second approach directly leads to an accurate solution to the task of planning a multifactorial multilevel experiment for an important special case where the costs of implementing the experiment plan are proportional to the total number of single-level transitions performed by all factors. At the same time, it has been proven that the proposed procedure for forming a route that implements the experiment plan minimizes the total number of one-level changes in the values of factors. Examples of problem solving are given

Stably continuous frames

1988 ◽  
Vol 104 (1) ◽  
pp. 7-19 ◽  
Author(s):  
B. Banaschewski ◽  
G. C. L. Brummer
Keyword(s):  
Lattice Theory ◽  
The Other ◽  
Hausdorff Spaces ◽  
Important Special Case ◽  
Compact Spaces ◽  
The One ◽  
Continuous Frames ◽  
Special Case

In the lattice theory that underlies topology, that is, in the study of frames, a class of frames arising naturally is that of the stably continuous frames (see §0 for definitions). On the one hand, they correspond to the most reasonable not necessarily Hausdorff compact spaces, and on the other, they are precisely the retracts of coherent frames. Moreover, an important special case of stably continuous frames are the compact regular frames which correspond to compact Hausdorff spaces.

MEMBRANE EQUILIBRIA, THE GIBBS–DALTON LAW AND THE ENTROPY OF MIXING

1949 ◽  
Vol 27b (12) ◽  
pp. 988-1014
Author(s):  
E. A. Flood ◽  
G. C. Benson
Keyword(s):  
The Other ◽  
Explicit Solutions ◽  
Classical Thermodynamic ◽  
Equilibrium Conditions ◽  
Semipermeable Membranes ◽  
Equilibrium Pressure ◽  
Entropy Of Mixing ◽  
Ideal Gases ◽  
Special Case ◽  
Thermodynamic Basis

When two pure fluids whose pressures are p1e and p2e, respectively, are separated by means of semipermeable membranes from a mixture of these fluids, and under equilibrium conditions the pressure of the mixture is P, then the net pressures sustained by the diaphragms are P − p1e and P − p2e respectively. The assumption that these net pressures are p2e and p1e, respectively, is equivalent to assuming the Gibbs–Dalton law, namely p1e + p2e = P. It is shown that the Gibbs–Dalton law when applied to fluids that are not ideal gases leads to consequences which are entirely contrary to experience and that as applied to ideal gases it has neither an experimental nor theoretical thermodynamic basis. It is shown that the Gibbs–Dalton law is only a special case of Dalton's law and that the classical thermodynamic paradox in the entropy of mixing of ideal gases is based on the erroneous assumption that the Gibbs–Dalton law necessarily holds when Dalton's law holds. It is shown that when two ideal gases obey Dalton's law of mixing, it is thermodynamically quite possible for the equilibrium pressure of one pure gas to be increased while that of the other is decreased, as well as the more familiar case of "chemical reaction", where the equilibrium pressures of both are decreased. It is shown that there is no purely thermodynamic requirement that different kinds of molecules in mixtures of ideal gases shall have the same mean translatory kinetic energy. The ideas underlying membrane equilibria are discussed in some detail. Some general condition equations which must be met are given, together with a few explicit solutions of these equations for special simple cases.

scholarly journals On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

2021 ◽  
Author(s):  
Olivier Ozenda ◽  
Epifanio G. Virga
Keyword(s):  
Free Energy ◽  
Dimension Reduction ◽  
Three Dimensional ◽  
Energy Functional ◽  
The Other ◽  
Variational Model ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Kinematic Constraint ◽  
Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

scholarly journals The game of Double-Silver on intervals

2001 ◽  
Vol 26 (8) ◽  
pp. 485-496 ◽  
Author(s):  
Gerald A. Heuer
Keyword(s):  
Measure Zero ◽  
Optimal Strategies ◽  
The Other ◽  
Parameter Plane ◽  
Left Endpoint ◽  
Relative Placement ◽  
Special Case

Silverman's game on intervals was analyzed in a special case by Evans, and later more extensively by Heuer and Leopold-Wildburger, who found that optimal strategies exist (and gave them) quite generally when the intervals have no endpoints in common. They exist in about half the parameter plane when the intervals have a left endpoint or a right endpoint, but not both, in common, and (as Evans had earlier found) exist only on a set of measure zero in this plane if the intervals are identical. The game of Double-Silver, where each player has its own threshold and penalty, is examined. There are several combinations of conditions on relative placement of the intervals, the thresholds and penalties under which optimal strategies exist and are found. The indications are that in the other cases no optimal strategies exist.

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

2004 ◽  
Vol 04 (01) ◽  
pp. 63-76 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Hausdorff Dimension ◽  
Finite Subset ◽  
Continued Fraction ◽  
Cantor Set ◽  
Computational Method ◽  
Accurate Approximation ◽  
Natural Numbers ◽  
Important Special Case ◽  
The One ◽  
Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

Silent Victims Revisited: The Special Case of Domestic Violence

PEDIATRICS ◽  
1995 ◽  
Vol 96 (3) ◽  
pp. 511-513
Author(s):  
Barry Zuckerman ◽  
Marilyn Augustyn ◽  
Betsy McAlister Groves ◽  
Steven Parker
Keyword(s):  
Domestic Violence ◽  
Community Violence ◽  
Clinical Experience ◽  
Behavioral Problems ◽  
The Other ◽  
Media Attention ◽  
Witness Violence ◽  
Primary Focus ◽  
Special Case ◽  
At Home

In a commentary published previously, we communicated our concern regarding the plight of children who witness violence.1 Research suggests that children who witness violence suffer significant psychologic and behavioral problems that interfere with their ability to function in school, at home, and with peers. The primary focus of that commentary was children who witnessed community violence. Our ongoing clinical experience, heightened by media attention on domestic violence, including the O.J. Simpson case, leads us to revisit silent victims with a sole focus on those children who witness domestic violence. Domestic violence is a particularly devastating event for a child who, in the presence of danger, typically turns to a parent for protection and for whom there is no comfort or security if one parent is the perpetrator of violence, and the other is a terrified victim.

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

Existence of Periodic Solution for Equation of Motion of Simple Beams With Harmonically Variable Length

1995 ◽  
Author(s):  
Ebrahim Esmailzadeh ◽  
Gholamreza Nakhaie-Jazar ◽  
Bahman Mehri
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Nonlinear Function ◽  
Axial Direction ◽  
Equation Of Motion ◽  
The Other ◽  
Sufficient Condition ◽  
Vibrating String ◽  
The Stability ◽  
Special Case

Abstract The transverse vibrating motion of a simple beam with one end fixed while driven harmonically along its axial direction from the other end is investigated. For a special case of zero value for the rigidity of the beam, the system reduces to that of a vibrating string with the corresponding equation of its motion. The sufficient condition for the periodic solution of the beam is then derived by means of the Green’s function and Schauder’s fixed point theorem. The criteria for the stability of the system is well defined and the condition for which the performance of the beam behaves as a nonlinear function is stated.

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